#507305

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,37)(32,43)(33,41)(34,44)(35,42)(36,38)(39,45)(40,46)(47,48);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,30,45,36,20,34,47,21,31,22,32,23,33,46,35)(18,24,37,43,38,41,44,42,26,29,27,39,28,25,40,48);(10,14)(13,15)(17,20,21,33)(18,25,26,41)(19,23,31,36)(22,32,30,45)(24,38,29,28)(27,43,37,39)(34,35,46,47)(40,48,44,42);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,26)(24,29)(25,41)(27,37)(28,38)(39,43)(40,44)(42,48)

Polycyclic presentation:

2^9|b^2=fgi,c^2=i,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414501);